By Thomas Ernst

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. typically, the cumbersome Gasper-Rahman notation was once used, however the released works on q-calculus regarded assorted looking on the place and through whom they have been written. This confusion of tongues not just advanced the theoretical improvement but in addition contributed to q-calculus closing a ignored mathematical box. This booklet overcomes those difficulties through introducing a brand new and engaging notation for q-calculus in accordance with logarithms.For example, q-hypergeometric services are actually visually transparent and straightforward to track again to their hypergeometric mom and dad. With this new notation it's also effortless to determine the relationship among q-hypergeometric services and the q-gamma functionality, anything that in the past has been overlooked.

The booklet covers many issues on q-calculus, together with exact features, combinatorics, and q-difference equations. except a radical overview of the historic improvement of q-calculus, this booklet additionally provides the domain names of contemporary physics for which q-calculus is appropriate, equivalent to particle physics and supersymmetry, to call only a few.

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**Example text**

Dyson and Ernest Barnes (1874–1953). Interestingly enough, all of these turned into special functions. Barnes became a bishop and wrote a long series of papers on special functions and difference calculus. In a way, Barnes was a predecessor of Nørlund, but they studied different problems. Although Barnes was an excellent mathematician, far better than Jackson, he never returned to the academia after a few years as a teacher. In his youth, Bailey met Jackson in the Navy and they certainly discussed q-calculus already then.

Br (a1 ) · · · (ap ) . (b1 ) · · · (br ) The meter, the measure unit for the length, can be compared with the tion, a basis for q-calculus. 44) q func- q function is defined by ⎧ 1−z , ∞ ⎨ 1;q z;q ∞ (1 − q) q (z) ≡ ⎩ 1;q −1 ∞ (q − 1)1−x q (x2) , x;q −1 q ∞ if 0 < |q| < 1, if |q| > 1. 45) Here we deviate from the usual convention q < 1, because we want to work with meromorphic functions of several variables. The reason is that the q-analogue of the Euler reflection formula involves the first Jacobi theta function, which by construction is a complex function, not only real.

30) above. 3 Finite differences and Bernoulli numbers 35 3. Yves André [22, p. 685] uses a notation that is equivalent to a; q n . André [22, p. 692] also denotes Dq by δσ . Indian: V. Rajeswari and K. Srinivasa Rao in 1991 [426] and in 1993 [470, p. 72] use my umbral notation in connection with the q-analogues of the 3-j and 6-j coefficients. Russian: 1. Also Gelfand has used a similar notation in one of his few papers [228, p. 38] on q-calculus. 31) for the q-hypergeometric series. If all αi and βi are non-zero, it is convenient to pass to the new parameters ai , bi , where αi = q ai , βi = q bi .